A340107 - OEIS

%I #19 Sep 11 2022 09:29:56

%S 1,1,1,1,2,2,1,3,6,0,1,4,12,16,16,1,5,20,50,90,80,1,6,30,108,300,552,

%T 516,1,7,42,196,742,2100,3990,3794,1,8,56,320,1536,5888,16976,32656,

%U 31456,1,9,72,486,2826,13680,53046,154350,299628,290970,1,10,90,700,4780,27960,136380,532340,1559040,3044900,2974380

%N Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive.

%C In a convex n-gon, the number of paths using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.

%F T(n,k) = n*(A340106(n-1,k-1) - S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0. [exception added by _Xiangyu Chen_, Aug 19 2022]

%e n\k   0     1     2     3     4     5     6     7     8

%e 0     1

%e 1     1     1

%e 2     1     2      2

%e 3     1     3      6      0

%e 4     1     4     12     16     16

%e 5     1     5     20     50     90    80

%e 6     1     6     30    108    300   552    516

%e 7     1     7     42    196    742  2100   3990  3794

%e 8     1     8     56    320   1536  5888  16976 32656 31456

%Y Cf. A338526, A338838, A338849.

%Y Cf. A340106, A340108.

%K nonn,tabl

%O 0,5

%A _Xiangyu Chen_, Dec 28 2020